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This paper presents a review and critical analysis of the mathematical literature concerning the modeling of vehicular traffic and crowd phenomena. The survey of models deals with the representation scales and the mathematical frameworks that are used for the modeling approach. The paper also considers the challenging objective of modeling complex systems consisting of large systems of individuals interacting in a nonlinear manner, where one of the modeling difficulties is the fact that these systems are difficult to model at a global level when based only on the description of the dynamics of individual elements. The review is concluded with a critical analysis focused on research perspectives that consider the development of a unified modeling strategy. [ABSTRACT FROM AUTHOR]

We extend the classical Lighthill–Whitham–Richards (LWR) traffic model allowing different maximal speeds for different vehicles. Then we add a uniform bound on the traffic speed. The result, presented in this paper, is a new macroscopic model displaying two phases based on a nonsmooth 2 × 2 system of conservation laws. This model is compared with other models of the same type in the current literature, as well as with a kinetic one. Moreover, we establish a rigorous connection between a microscopic follow-the-leader model based on ordinary differential equations and this macroscopic continuum model. [ABSTRACT FROM AUTHOR]

In the present paper a kinetic model for vehicular traffic leading to multivalued fundamental diagrams is developed and investigated in detail. For this model phase transitions can appear depending on the local density and velocity of the flow. A derivation of associated macroscopic traffic flow equations from the kinetic equation is given. Moreover, numerical experiments show the appearance of stop and go waves for highway traffic with a bottleneck. [ABSTRACT FROM AUTHOR]